Let F 0 = {f i } i∈In 0 be a finite sequence of vectors in C d and let a = (a i ) i∈I k be a finite sequence of positive numbers. We consider the completions of F 0 of the form F = (F 0 , G) obtained by appending a sequence G = {g i } i∈I k of vectors in C d such that g i 2 = a i for i ∈ I k , and endow the set of completions with the metric d(F ,F ) = max{ g i −g i : i ∈ I k } wherẽ F = (F 0 ,G). In this context we show that local minimizers on the set of completions of a convex potential P ϕ , induced by a strictly convex function ϕ, are also global minimizers. In case that ϕ(x) = x 2 then P ϕ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn's conjecture on the FOD.

Lidskii's additive inequalities (both for eigenvalues and singular values) can be interpreted as an explicit description of global minimizers of functions that are built on unitarily invariant norms, with domains consisting of certain orbits of matrices (under the action of the unitary group). In this paper, we show that Lidskii's inequalities actually describe all global minimizers of such functions and that local minimizers are also global minimizers. We use these results to obtain partial results related to local minimizers of generalized frame operator distances in the context of finite frame theory.

Let S ∈ M d (C) + be a positive semidefinite d × d complex matrix and let a = (a i ) i∈I k ∈ R k >0 , indexed by I k = {1, . . . , k}, be a k-tuple of positive numbers. Let T d (a) denote the set of familiesis the product of spheres in C d endowed with the product metric. For a strictly convex unitarily invariant norm N in M d (C), we consider the generalized frame operator distance function Θ (N , S , a) defined on T d (a), given byIn this paper we determine the geometrical and spectral structure of local minimizers G 0 ∈ T d (a) of Θ (N , S , a) . In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of N .

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